Question

Prove that every quadratic function is either always concave up or always concave down.

Short answer

Proved that every quadratic function is either always concave up or always concave down.

Step-by-step solution

Step 1. Given Information.

Given a quadratic function. Let that function be$f\left(x\right)=a{x}^2+bx+c.$

Step 2. Theorem.

The Derivative Measures Where a Function is Increasing or Decreasing

Let f be a function that is differentiable on an interval I.

(a) If $f\text{'}$is positive in the interior of I, then f is increasing on I.

(b) If $f\text{'}$is negative in the interior of I, then f is decreasing on I.

(c) If $f\text{'}$is zero in the interior of I, then f is constant on I.

Step 3. Proof.

$$\begin{gathered} f\left(x\right)=a{x}^2+bx+c, \\ Differentiatingbothsidesw.r.txweget, \\ f\text{'}\left(x\right)=2ax+b, \\ Differentiatingbothsidesw.r.txonceagainweget, \\ f\text{'}\text{'}\left(x\right)=2a, \\ Now,f\text{'}\text{'}\left(x\right)isacons\tan tanddoesnotdependonxandaccordingtothe \\ valueofcons\tan taitwillbeeitherpositiveornegativeandaccordinglythe \\ functionf\left(x\right)willbeconcaveupwardorconcavedownwardrespectively. \end{gathered}$$

Hence, Proved.